Efficient computation of the volume of a polytope in high-dimensions using Piecewise Deterministic Markov Processes

02/18/2022
by   Augustin Chevallier, et al.
7

Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to the polytope, using e.g. Hamiltonian Monte Carlo. We present a new sampling strategy that uses a Piecewise Deterministic Markov Process. Like Hamiltonian Monte Carlo, this new method involves simulating trajectories of a non-reversible process and inherits similar good mixing properties. However, importantly, the process can be simulated more easily due to its piecewise linear trajectories - and this leads to a reduction of the computational cost by a factor of the dimension of the space. Our experiments indicate that our method is numerically robust and is one order of magnitude faster (or better) than existing methods using Hamiltonian Monte Carlo. On a single core processor, we report computational time of a few minutes up to dimension 500.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

08/26/2018

Hypercoercivity of Piecewise Deterministic Markov Process-Monte Carlo

In this paper we derive spectral gap estimates for several Piecewise Det...
06/09/2022

Randomized Time Riemannian Manifold Hamiltonian Monte Carlo

In the last decade several sampling methods have been proposed which rel...
12/24/2021

Concave-Convex PDMP-based sampling

Recently non-reversible samplers based on simulating piecewise determini...
07/14/2020

An algorithm for estimating volumes and other integrals in n dimensions

The computational cost in evaluation of the volume of a body using numer...
08/26/2018

Hypocoercivity of Piecewise Deterministic Markov Process-Monte Carlo

In this work, we establish L^2-exponential convergence for a broad class...
02/26/2022

Metropolis Adjusted Langevin Trajectories: a robust alternative to Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a widely used sampler, known for its ef...
05/14/2019

Practical Volume Estimation by a New Annealing Schedule for Cooling Convex Bodies

We study the problem of estimating the volume of convex polytopes, focus...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.